Barack Obama today rejected calls to abandon his Afghan war strategy and either offer an open-ended commitment to US troops fighting there or start withdrawing immediately, after his dismissal of the US and Nato commander in Kabul, General Stanley McChrystal.

Leading Republican politicians and the former American secretary of state Henry Kissinger have called on the president to drop a July 2011 deadline to begin withdrawing US troops, saying that it undermines the effort to defeat the Taliban.

Voices on the Democratic party's left want withdrawal to begin immediately, saying the war cannot be won.

But today, Obama said he intended to stick with the strategy and timetable agreed last year, while indicating that US troops could remain in Afghanistan in significant numbers well after the withdrawal is due to start next summer.

"We did not say, starting in July 2011, suddenly there will be no troops from the United States or allied countries in Afghanistan," Obama said at a press conference with the Russian president, Dmitry Medvedev, who sidestepped a question about whether, in light of the Soviet Union's defeat in Afghanistan, a foreign army can expect to win a war in Afghanistan.

"We didn't say we'd be switching off the lights and closing the door behind us. We said we'd begin a transition phase that would allow the Afghan government to take more and more responsibility."

Obama added that part of the strategy would include a reassessment at the end of this year. "In December of this year, a year after the strategy has been put in place, at a time when the additional troops have been in place and have begun implementing the strategy, then we'll conduct a review and make an assessment. Is the strategy working? Is it working in part? Are there other aspects of it that aren't working?" he said.

Lindsey Graham, a Republican senator who sits on the powerful armed services committee and backed Obama's dismissal of McChrystal, said the July 2011 deadline undercut the war effort: "It empowers our enemies. It confuses our friends. And I think it needs to be re-evaluated."

Graham said McChrystal's replacement, General David Petraeus, who led the US troop surge in Iraq, had testified to Congress that he would urge Obama to delay the pullout if he believed it was unwise.

"If the president says, no matter what General Petraeus may recommend, we're going to leave in July of 2011, we will lose the war," Graham said.

Kissinger, writing in the Washington Post, warned of the potential for a collapse in US public support for the conflict – similar to that which occurred during the Vietnam conflict – that could lead to a political focus on "an exit strategy with the emphasis on exit, not strategy". He said it was a mistake to impose a deadline for US involvement.

"The central premise is that, at some early point, the United States will be able to turn over security responsibilities to an Afghan government and national army whose writ is running across the entire country. This turnover is to begin next summer. Neither the premise nor the deadline is realistic," he wrote.

But a Democratic senator, Carl Levin, the chairman of the armed services committee, called for the US to rapidly turn over responsibility for military operations to the Afghans. "There's a greater chance of success if people in Kandahar, in Afghanistan, see it's their army that is in control. It is their army that is taking the lead against the Taliban, rather than outsiders."

Other critics of the war have noted that the article in Rolling Stone article magazine that forced McChrystal out offered a disturbing insight into the mindset of America's frontline troops in Afghanistan. The troops are quoted as saying that the ousted general's strategy constrained their ability to fight the Taliban, and that they are losing the war.

In Kabul, there is a belief that changes at the top of the International Security Assistance Force (Isaf) are inevitable, including the purging of McChrystal's entourage, especially the source of the anonymous quotes in Rolling Stone magazine criticising senior US officials .

Mark Sedwill, Nato's ambassador in Kabul, who worked closely with McChrystal, said the former commander of ISAF was "an outstanding military commander". He said: "But the mission is bigger than one man and, in choosing General Petraeus, [Obama] has picked one of the absolutely finest soldiers in the world ... and exactly the right man to lead this campaign over the coming period."

Petraeus is regarded as a good choice by those who will have to work with him because of his success in leading the "surge" in Iraq.

But even the strategy's strongest supporters say it is a task far harder than Petraeus faced in Baghdad. His main enemy is an unforgiving political timetable.

The scale and timing of the drawdown will depend on how the campaign is going. If violence has decreased, it may be possible to "slow the Washington" clock, as McChrystal's team used to say.

If things continue to look like they are going disastrously – and if this September's Afghan parliamentary elections are a repeat of last year's electoral fiasco – policymakers in Washington are likely to conclude the strategy is unworkable and will look for alternative ways of dealing with Afghanistan, almost certainly involving far fewer troops concentrating mostly on counter-terrorism operations against al-Qaida.

Showing progress will be tough. Petraeus will have to demonstrate clear declines in insurgent violence in the most violent areas of southern Afghanistan, particularly in Marjah, a small farming community in Helmand that was the target of a huge Isaf offensive in February.

But progress in Helmand has been frustratingly slow, and in Kandahar, the other main target of the strategy Petraeus is inheriting, it has been delayed even though it should already have begun in earnest.

Petraeus will need to prove that strides are being made on "transition" – getting Afghanistan's corrupt, poorly trained and undermanned army and police force in a position to take over security.

He will also need to have shown progress on country's "reintegration" attempt to persuade mid and low-level insurgents to stop fighting and settle down in peace. But the plan's ambitious calls for huge numbers of jobs to be created for insurgents (who will also be offered vocational training) has been met with great scepticism by experts.

And though people talk of July 2011 as a "deadline", the actual cut-off point is more like November, when Nato foreign ministers meet in Lisbon. At around the same time there will be a US assessment of progress. It is assumed in Kabul that Obama will want to be in a position to decide the fate of US operations in Afghanistan by his state of the union address in January.